Problem 15 - Entrance Test

Given sin θ - cos θ = 0, which implies sin θ = cos θ. Since sin θ = cos θ, we can divide by cos θ (assuming cos θ ≠ 0) to get tan θ = 1. For acute angles, θ = 45°. Now, we need to find the value of sin⁴ θ + cos⁴ θ. Substitute θ = 45°: sin⁴ 45° + cos⁴ 45° = (sin 45°)⁴ + (cos 45°)⁴ = (1/√2)⁴ + (1/√2)⁴ = (1/2)² + (1/2)² = 1/4 + 1/4 = 2/4 = 1/2. Alternatively, from sin θ = cos θ, square both sides to get sin² θ = cos² θ. We also know sin² θ + cos² θ = 1. Substitute cos² θ = sin² θ into the identity: sin² θ + sin² θ = 1. 2 sin² θ = 1, so sin² θ = 1/2. Since cos² θ = sin² θ, then cos² θ = 1/2. Now, sin⁴ θ + cos⁴ θ = (sin² θ)² + (cos² θ)² = (1/2)² + (1/2)² = 1/4 + 1/4 = 1/2.